Multiple disjointness and invariant measures on minimal distal flows

Juho Rautio. "Multiple disjointness and invariant measures on minimal distal flows." Studia Mathematica 228.2 (2015): 153-175. <http://eudml.org/doc/285847>.

@article{JuhoRautio2015,
abstract = {We examine multiple disjointness of minimal flows, that is, we find conditions under which the product of a collection of minimal flows is itself minimal. Our main theorem states that, for a collection $\{X_\{i\}\}_\{i∈I\}$ of minimal flows with a common phase group, assuming each flow satisfies certain structural and algebraic conditions, the product $∏_\{i∈I\} X_\{i\}$ is minimal if and only if $∏_\{i∈I\} X_\{i\}^\{eq\}$ is minimal, where $X_\{i\}^\{eq\}$ is the maximal equicontinuous factor of $X_\{i\}$. Most importantly, this result holds when each $X_\{i\}$ is distal. When the phase group T is ℤ or ℝ, we can apply this idea to construct large minimal distal product flows with many ergodic measures. We determine the exact cardinality of (ergodic) invariant measures on the universal minimal distal T-flow. Equivalently, we determine the cardinality of (extreme) invariant means on (T), the space of distal functions on T. This cardinality is $2^\{\}$ for both ergodic and invariant measures. The size of the quotient of (T) by a closed subspace with a unique invariant mean is found to be non-separable by using the same techniques.},
author = {Juho Rautio},
journal = {Studia Mathematica},
keywords = {PI flow; distal flow; maximal equicontinuous factor; disjointness; invariant measure; invariant Mean},
language = {eng},
number = {2},
pages = {153-175},
title = {Multiple disjointness and invariant measures on minimal distal flows},
url = {http://eudml.org/doc/285847},
volume = {228},
year = {2015},
}

TY - JOUR
AU - Juho Rautio
TI - Multiple disjointness and invariant measures on minimal distal flows
JO - Studia Mathematica
PY - 2015
VL - 228
IS - 2
SP - 153
EP - 175
AB - We examine multiple disjointness of minimal flows, that is, we find conditions under which the product of a collection of minimal flows is itself minimal. Our main theorem states that, for a collection ${X_{i}}_{i∈I}$ of minimal flows with a common phase group, assuming each flow satisfies certain structural and algebraic conditions, the product $∏_{i∈I} X_{i}$ is minimal if and only if $∏_{i∈I} X_{i}^{eq}$ is minimal, where $X_{i}^{eq}$ is the maximal equicontinuous factor of $X_{i}$. Most importantly, this result holds when each $X_{i}$ is distal. When the phase group T is ℤ or ℝ, we can apply this idea to construct large minimal distal product flows with many ergodic measures. We determine the exact cardinality of (ergodic) invariant measures on the universal minimal distal T-flow. Equivalently, we determine the cardinality of (extreme) invariant means on (T), the space of distal functions on T. This cardinality is $2^{}$ for both ergodic and invariant measures. The size of the quotient of (T) by a closed subspace with a unique invariant mean is found to be non-separable by using the same techniques.
LA - eng
KW - PI flow; distal flow; maximal equicontinuous factor; disjointness; invariant measure; invariant Mean
UR - http://eudml.org/doc/285847
ER -